Not all Ironborn are the same. Their are ironborn who supported Asha. Asha wanted peace. Not to mention Ironborn have no fertile ground to grow. The Starks rape, and pillage the Riverlands. Why don't people hate the Starks? The Targaryens, Lannisters and Ironborn are mild compared to other houses.

There are 5 reason why he gave the orders.
1). Show loyalty to Robert.
2). Get Cersei married to Robert.
3). Get revenge on the Targaryens.
4). Get revenge on Rhaegar for not keeping his word to marry Cersei (The Defiance of Duskendale).
5). Get Lannister blood on the throne.

Tywin did it to show loyalty to Robert. If he didn't Robert would be at war with the Lannisters. Rhaegar married Lyanna. Making Elia and his children disinherited. Even if the kids survived. If anyone found out the marriage happened. Elia's kids wouldn't inherit the throne.

Tyrion was born similar how Dany's child was born. Tywin knew Tyrion wasn't his kid. He knew the King took liberties with his wife.
http://awoiaf.westeros.org/images/f/f1/Morgaine_le_Fee_Rhaego_Targaryen.jpg